A Kadison-Dubois representation for associative rings.
The Kadison-Dubois representation has a long and remarkable history. It appeared in many settings and at different levels of generalization. Its generalization to the case of commutative rings was found by \textit{T. Jacobi} [his Dissertation paper and Math. Z. 237, No. 2, 259-273 (2001; Zbl 0988.54039)] and then extended by \textit{M. Marshall} [Math. Z. 242, No. 2, 217-225 (2002; Zbl 1060.13008)]. The aim of this paper is to generalize Jacobi's version of the Kadison-Dubois representation to associative rings. The author finds necessary and sufficient conditions for a module to satisfy the so called Kadison-Dubois property (KDP for short). For a preprime \(T\) and \(T\)-module \(M\) in a unital associative ring \(A\) he proves the equivalence of the following statements: (1) \(M\) satisfies the KDP, (2) \(M\) is Archimedean and its infinitesimal hull \(\text{Arch}(M)\) is closed under multiplication, (3) \(\text{Arch}(M)\) is an Archimedean preprime. This criterion is used to obtain the main result: if \(T\) is a preordering and \(M\) is an Archimedean \(T\)-module in a unital associative ring \(A\), then \(M\) satisfies the KDP.
scientific article; zbMATH DE number 2078859
scientific article